Abstract

We consider the application of tomography to the reconstruction of 2-D vector fields. The most convenient sensor configuration in such problems is the regular positioning along the domain boundary. However, the most accurate reconstructions are obtained by sampling uniformly the Radon parameter domain rather than the border of the reconstruction domain. This dictates a prohibitively large number of sensors and impractical sensor positioning. In this paper, we propose uniform placement of the sensors along the boundary of the reconstruction domain and interpolation of the measurements for the positions that correspond to uniform sampling in the Radon domain. We demonstrate that when the cubic spline interpolation method is used, a 60 times reduction in the number of sensors may be achieved with only about 10% increase in the error with which the vector field is estimated. The reconstruction error by using the same sensors and ignoring the necessity of uniform sampling in the Radon domain is in fact higher by about 30%. The effects of noise are also examined.

Figures (11)

Tracing line AB unites two virtual sensors that reside at points A and B. The tracing line is defined by the two parameters ρ and θ (Radon domain coordinates) and goes through the digitized square reconstruction region of size 2U×2U. The line segment is sampled with sampling step Δs. The angle between the line segment and the positive direction of the x-axis is w. The size of the tiles with which we sample the 2-D space is P×P. Also shown is the unit vector ŝ, which is parallel to line segment AB.

Comparison of the reconstruction performance for the cases when reconstruction was based on: (i) line-integral data from regularly placed sensors (RS) in relation to (x,y) coordinates; (ii) interpolated line-integral data obtained at virtual sensors that corresponded to uniform sampling of the Radon space and the employed interpolation method was the 1-D linear (IP1), the 1-D piecewise cubic spline (IP2), the piecewise cubic Hermite (IP3), the bilinear (IP4), the bicubic (IP5), and the 2-D spline (IP6); (iii) uniform sampling (US) of the parameter space using the actual measurements. The location of the source of the electric field was at (19, −19).

Simulation results when the location of the source of the electric field was (from top to bottom) at (19, −19), (−16, 21), (−21,−12), and (24, 14.5): (a) the recovered vector field when reconstruction was based on interpolated line-integral data (1-D piecewise cubic spline method) obtained at virtual sensors that corresponded to uniform sampling of the Radon space with Δρ=0.5 and Δθ=1.5°; (b) the theoretical electric field as computed from Coulomb’s law.

Comparison of the reconstruction performance in noisy environments for the cases: (i) when integral data from regularly placed sensors were used; (ii) when interpolated measurements that corresponded to uniform sampling with Δρ=0.5 and Δθ=1.5° were used; (iii) when actual measurements that corresponded to uniform sampling with Δρ=0.5 and Δθ=1.5° were used. (a), (b) Errors in vector field orientation and magnitude when noise was added to the measurements of 25% of the sensors, as a percentage of the true value. (c), (d) Errors in vector field orientation and magnitude when small perturbations in the sensor positions were added. Position perturbations were a percentage of the true positions. (e), (f) Errors in vector field orientation and magnitude when both sensors’ measurements and positions were changed by a percentage of their true values. In all cases, 25% of the sensors were perturbed.

Comparison of the reconstruction performance in noisy environments for the cases: (i) when integral data from regularly placed sensors were used; (ii) when interpolated (1-D piecewise cubic spline) measurements that corresponded to uniform sampling with Δρ=0.5 and Δθ=1.5° were used; and (iii) when actual measurements that corresponded to uniform sampling with Δρ=0.5 and Δθ=1.5° were used. (a) Error in vector field orientation and (b) error in magnitude when Gaussian noise of zero mean was added to the measurements of the sensors.